Through the ages mathematicians have been puzzled and thrilled; puzzled with the secret of numbers and thrilled with the infinite possibilities that the science of mathematics has to offer. From time to time, new ways of using numbers and new numbering system are discovered. Such discoveries reveal a new infinity of possibilities that can be mind boggling, since in mathematics or in any other science for that matter, everything is in the hidden to be discovered, analyzed and expanded.
In the field of encryption, new algorithms are found and used in the science of the ability to cipher and decipher information with the use of mathematical formulas and to some extent they all use the science of the shadow numbers. There are two schemes of encryption algorithm, symmetric (private) and asymmetric (public).
The symmetric scheme uses a single key called the private key and it is used both, to encrypt and decrypt. The private key must be kept private all the time that is, kept secret, since only one key is used and anyone in possession of it will be able to cipher (encrypt) and decipher (decrypt) the message that is associated with it.
The asymmetric scheme is when two keys are used, one for enciphering and the other for deciphering the content, there is, the public and the private encryption-keys pair. The public key as its name implies, it is to be used by anyone who comes across it and it works in conjunction with its equivalent private key. The public key is used for enciphering the content and the private key equivalent of the public key is used for deciphering the enciphered content.
An asymmetric scheme has advantages: it can be viewed as a two-way lane one for each direction, that is, the private key can be used for enciphering as well and the public key equivalent to the private key for deciphering what was enciphered with the private key. The private key can encipher content to a group of recipients and everyone in possession of the public key equivalent to the private key can decipher it. This process happens when the sender—holder of the private—sends a message to the group having the public key.
In general, the asymmetric scheme is slower than its symmetric counterpart. In a great number of situations, a combination of both schemes is used for the purpose of security and speed. The symmetric scheme is used to encipher the content and the asymmetric one is used to encipher the content's key. In this way, the best of the two worlds can be achieved. The symmetric scheme encrypts the content and produces the content's key and the asymmetric scheme encrypts the content's key.
The asymmetric scheme involves mathematical formulas and in most cases employs numerical exponentiations, which requires a great deal of computation power on both ends, for enciphering and deciphering. The asymmetric scheme works by providing two or more formulas for the creation of the two-key combination, for enciphering and deciphering. The two-key pair and the two keys as originating numbers, produce mathematical values equivalent to each other. These equivalents are considered the shadows of the originating numbers.
Hellman, et al., U.S. Pat. No. 4,200,770 (Hellman) teaches a cryptographic system that transmits a computationally secure cryptogram over an insecure communication channel without prearrangement of a cipher key. The conversers from transformations of exchanged transformed signals generate a secure cipher key. The conversers each possess a secret signal and exchange an initial transformation of the secret signal with the other converser. The received transformation of the other converser's secret signal is again transformed with the receiving converser's secret signal to generate a secure cipher key. The transformations use non-secret operations that are easily performed but extremely difficult to invert. It is infeasible for an eavesdropper to invert the initial transformation to obtain either converser's secret signal, or duplicate the latter transformation to obtain the secure cipher key.
Hellman teaches a cryptographic apparatus where two parties can safely exchange secured data through insecure channel without prior knowledge of the parties-common secret key. Hellman fails to teach, however, a common denominator in deriving the cryptographic keys without a laborious and expensive means for deriving the large-prime numbers values.
There are other means of encryption algorithm that include two keys: a private key and a public key. The intended recipient of the ciphered text that is, the encrypted text, only knows the public key by the private key equivalent. One of the most popular public key algorithms is the RSA algorithm, named after its three inventors—Ron Rivest, Adi Shamir, and Leonard Adleman. A message M is encrypted using the formula C=ME mod N, where N is the product of two large primes numbers P. The exponent E is a number relatively prime to (P−1)(Q−1). Q is chosen at random. The encrypted message C is deciphered using the formula M=CD mod N where D=E−1 mod((p−1)(q−1)). The exponent E and modulus N are used as the public key. The exponent D is the private key. The primes P and Q are not needed once the public and private keys have been computed but should remain secret.